8.1: Non-Integer Powers as Multi-Valued Operations
Given a complex number in its polar representation, \(z = r\exp[i\theta]\) , raising to the power of \(p\) could be handled this way: \[z^p = \left(re^{i\theta}\right)^p = r^p e^{ip\theta}.\] Let’s take a closer look at the complex exponential term \(e^{ip\theta}\) . Since \(\theta = \mathrm{arg}(z)\) is an angle, we can change it by any integer multiple of \(2\pi\) without altering the value of \(z\) . Taking this fact into account, we can re-write the above equation more carefully as \[z^p = \left(r\,e^{i(\theta + 2\pi n)}\right)^p = \left(r^p e^{ip\theta} \right) e^{2\pi i n p} \quad\; \mathrm{where}\;\; n\in\mathbb{Z}.\] Thus, there is an ambiguous factor of \(\exp(2\pi i n p)\) , where \(n\) is any integer. If \(p\) is an integer, there is no problem, since \(2\pi n p\) will be an integer multiple of \(2\pi\) , so \(z^p\) has the same value regardless of \(n\) : \[z^p = r^p e^{ip\theta} \;\;\textrm{unambiguously} \;\;\;(\text{if}\,p\in\mathbb{Z}).\] But if \(p\) is not an integer, there is no unique answer, since \(\exp\left(2 \pi i np\right)\) has different values for different \(n\) . In that case, “raising to the power of \(p\) ” is a multi-valued operation . It cannot be treated as a function in the usual sense, since functions must have unambiguous outputs (see Chapter 0).
Roots of unity
Let’s take a closer look at the problematic exponential term, \[\exp\left(2\pi i np\right), \quad n \in \mathbb{Z}.\] If \(p\) is irrational, \(2\pi np\) never repeats itself modulo \(2\pi\) . Thus, \(z^p\) has an infinite set of values, one for each integer \(n\) .
More interesting is the case of a non-integer rational power, which can be written as \(p = P/Q\) where \(P\) and \(Q\) are integers with no common divisor. It can be proven using modular arithmetic (though we will not go into the details) that \(2\pi n\, (P/Q)\) has exactly \(Q\) unique values modulo \(2\pi\) : \[2\pi n\, \left(\frac{P}{Q}\right) = 2\pi \times \left\{0,\, \frac{1}{Q},\, \frac{2}{Q},\, \dots, \frac{(Q-1)}{Q} \right\} \quad(\mathrm{modulo} \; 2\pi).\] This set of values is independent of the numerator \(P\) , which merely affects the sequence in which the numbers are generated. We can clarify this using a few simple examples:
Example \(\PageIndex{1}\)
Consider the complex square root operation, \(z^{1/2}\) . If we write \(z\) in its polar respresentation, \[z = r e^{i\theta},\] then \[z^{1/2} = \left[r \, e^{i(\theta + 2 \pi n)} \right]^{1/2} = r^{1/2} \, e^{i\theta/2} \, e^{i \pi n}, \quad n \in \mathbb{Z}.\] The factor of \(e^{i\pi n}\) has two possible values: \(+1\) for even \(n\) , and \(-1\) for odd \(n\) . Hence, \[z^{1/2} = r^{1/2} \, e^{i\theta/2} \;\times\; \left\{1, -1\right\}.\]
Example \(\PageIndex{2}\)
Consider the cube root operation \(z^{1/3}\) . Again, we write \(z\) in its polar representation, and obtain \[z^{1/3} = r^{1/3} \, e^{i\theta/3} \, e^{2\pi i n/3}, \quad n \in \mathbb{Z}.\] The factor of \(\exp(2\pi i n/3)\) has the following values for different \(n\) : \[\nonumber\begin{array}{|c||c|c|c|c|c|c|c|c|c|} \hline n &\cdots & -2 & -1 & 0 & 1 & 2 & 3 & 4 & \cdots \\ \hline e^{2\pi i n/3} &\cdots & e^{2\pi i /3} & e^{-2\pi i /3} & \;\,\;1\;\,\; & e^{2\pi i /3} & e^{-2\pi i /3} & \;\,\;1\;\,\; & e^{2\pi i /3} & \cdots \\ \hline \end{array}\] From the pattern, we see that there are three possible values of the exponential factor: \[e^{2\pi i n/3} = \left\{1, e^{2\pi i /3}, e^{-2\pi i /3}\right\}.\] Therefore, the cube root operation has three distinct values: \[z^{1/3} = r^{1/3} \, e^{i\theta/3} \;\times\; \left\{1, e^{2\pi i /3}, e^{-2\pi i /3}\right\}.\]
Example \(\PageIndex{3}\)
Consider the operation \(z^{2/3}\) . Again, writing \(z\) in its polar representation, \[z^{2/3} = r^{2/3} \, e^{2i\theta/3} \, e^{4\pi i n/3}, \quad n \in \mathbb{Z}.\] The factor of \(\exp(4\pi i n/3)\) has the following values for different \(n\) : \[\nonumber\begin{array}{|c||c|c|c|c|c|c|c|c|c|} \hline n &\cdots & -2 & -1 & 0 & 1 & 2 & 3 & 4 & \cdots \\ \hline e^{4\pi i n/3} &\cdots & e^{-2\pi i /3} & e^{2\pi i /3} & \;\,\;1\;\,\; & e^{-2\pi i /3} & e^{2\pi i /3} & \;\,\;1\;\,\; & e^{-2\pi i /3} & \cdots \\ \hline \end{array}\] Hence, there are three possible values of this exponential factor, \[e^{2\pi i n (2/3)} = \left\{1, e^{2\pi i /3}, e^{-2\pi i /3}\right\}.\] Note that this is the exact same set we obtained for \(e^{2\pi i n/3}\) in the previous example, in agreement with the earlier assertion that the numerator \(P\) has no effect on the set of values. Thus, \[z^{2/3} = r^{2/3} \, e^{2i\theta/3} \;\times\; \left\{1, e^{2\pi i /3}, e^{-2\pi i /3}\right\}.\]
From the above examples, we deduce the following expression for rational powers: \[z^{P/Q} = r^{P/Q} \; e^{i\theta\, (P/Q)}\, \times \left\{1,\, e^{2\pi i \cdot (1/Q)},\, e^{2\pi i \cdot (2/Q)},\, \dots, e^{2\pi i \cdot [(1-Q)/Q]} \right\}.\] The quantities in the curly brackets are called the roots of unity . In the complex plane, they sit at \(Q\) evenly-spaced points on the unit circle, with \(1\) as one of the values:
Complex logarithms
Here is another way to think about non-integer powers. Recall what it means to raise a number to, say, the power of 5: we simply multiply the number by itself five times. What about raising a number to a non-integer power \(p\) ? For the real case, we used the following definition based on a combination of exponential and logarithm functions: \[x^p \equiv \exp\,\left[\,p\ln(x)\right].\] This definition relies on the fact that, for real inputs, the logarithm is a well-defined function. That, in turn, comes from the definition of the logarithm as the inverse of the exponential function. Since the real exponential is one-to-one, its inverse is also one-to-one.
The complex exponential, however, is many-to-one, since changing its input by any multiple of \(2\pi i\) yields the same output: \[\exp(z + 2\pi i n) = \exp(z) \cdot e^{2\pi i n} = \exp(z) \quad \forall\; n \in \mathbb{Z}.\] The inverse of the complex exponential is the complex logarithm . Since the complex exponential is many-to-one, the complex logarithm does not have a unique output. Instead, \(\ln(z)\) refers to an infinite discrete set of values, separated by integer multiples of \(2\pi i\) . We can express this state of affairs in the following way: \[\ln(z) = \big[\ln(z)\big]_{\mathrm{p.v.}}\; +\; 2 \pi i n, \quad n \in \mathbb{Z}. \label{logformula}\] Here, \([\ln(z)]_{\mathrm{p.v.}}\) denotes the principal value of \(\ln(z)\) , which refers to a reference value of the logarithm operation (which we’ll define later). Do not think of the principal value as the "actual" result of the \(\ln(z)\) operation! There are multiple values, each equally legitimate; the principal value is merely one of these possible results.
Plugging Eq. \(\eqref{logformula}\) into the formula \(z^p \equiv \exp\left[p\ln(z)\right]\) gives \[\begin{align} z^p &= \exp\Big\{p\big([\ln(z)]_{\mathrm{p.v.}} + 2\pi i n\big)\Big\}\\ &= \exp\Big\{p[\ln(z)]_{\mathrm{p.v.}}\Big\} \times e^{2\pi i np}, \quad n \in \mathbb{Z}.\end{align}\] The final factor, which is responsible for the multi-valuedness, are the roots of unity found in Section 8.1.